Geometric Problems

Sequences have wide applications in geometric problems, from simple figure dissection to complex fractal geometry.

Figure Dissection Problems

Example 1: Triangle Dissection

Connect the midpoints of the three sides of a triangle to obtain 4 small triangles. Repeat this process. How many triangles are there after the $n$th dissection?

Solution:

• 1st time: 4
• 2nd time: $4 \times 4 = 16$
• 3rd time: $4 \times 4 \times 4 = 64$
• $n$th time: $4^n$

This is a geometric sequence with common ratio 4.

Example 2: Square Dissection

Divide a square into four equal parts, remove the upper-right part, and repeat this process on the remaining three squares. What is the remaining area after the $n$th step?

Solution:

Let the area of the original square be 1. Each time $\frac{3}{4}$ remains:

$S_n = \left(\frac{3}{4}\right)^n$

This is a geometric sequence, and $\lim_{n \to \infty} S_n = 0$

Fractal Geometry

The Sierpinski Triangle

Start with an equilateral triangle and remove the small middle triangle each time, repeating infinitely.

After the $n$th step:

• Number of triangles: $3^n$
• Total area: $\left(\frac{3}{4}\right)^n S_0$ ($S_0$ is the original area)

As $n \to \infty$:

• The number of triangles $\to \infty$
• The total area $\to 0$

This is the wonder of fractals!

Infinite Series and Area

Example: An Infinitely Divided Square

A square with side length 1. Take away $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$ of its area in turn. What is the remaining area?

Solution:

The sum of the removed areas:

$S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1$

Remaining area: $1 - 1 = 0$

Solid Geometry

Example: Stacking Balls

Layer 1 has 1 ball, layer 2 has 4, layer 3 has 9, …, layer $n$ has $n^2$ balls. How many balls are there in the first $n$ layers in total?

Solution:

$S_n = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$

Practice Problems

Exercise 1

Divide a square with side length 1 into 4 small squares, then divide each small square into 4 even smaller squares, repeating this $n$ times. How many small squares are there after the $n$th step? What is the side length of each small square?

Exercise 2

An equilateral triangle with side length 1. Connect the midpoints of the sides to obtain 4 small triangles and remove the middle one. Repeat this operation on the remaining 3 triangles. Find the perimeter of the remaining figure after the $n$th operation.

Summary

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